The resulting solution \(t\mapsto \mathbf {1}_{D_t}\)  is called a vortex patch with initial patch \(\mathbf {1}_{D_0}.\)  Given any parametrization \(z(t,\cdot ):\mathbb {T}\rightarrow \partial D_t\)  of the boundary of the patch at time \(t,\)  it is well-known since the works [1]}, [2]} that for an active scalar equation, as in our case, the particles on the boundary move with the flow and remain at the boundary. Hence, at least in the smooth case, one can write the following equation called vortex patch equation.
\(\Big (\partial _tz(t,\theta )-\mathbf {v}\big (t,z(t,\theta )\big )\Big )\cdot \mathbf {n}\big (t,z(t,\theta )\big )=0,\) 
